Conduction cooling of a superconducting cable

ABSTRACT

A conduction-cooled superconducting power transmission cable wherein a High-Temperature Superconducting (HTS) wire is surrounded by an inner layer of thermal insulator, one or more layers of high thermal conductivity material, such as copper, and an outer layer of thermal insulator with cryogenic coolant sources distributed along the power transmission cable and coupled to the copper layers or both the copper layers and the HTS wire. The cryogenic coolant sources can be reservoirs, a distribution system of coolant or stand alone refrigeration systems. He H 2  or N 2  liquid coolant can be used. A method for calculating the parameters to maintain the critical temperature of the HTS wire and a method for calculating the cool down time from ambient conditions are disclosed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority from, and is a 35 U.S.C. § 111 (a)continuation of, co-pending PCT international application serial numberPCT/US2004/027220, filed on Aug. 20, 2004, which designates the U.S.,incorporated herein by reference in its entirety, which claims priorityfrom U.S. provisional application Ser. No. 60/497,163 filed on Aug. 22,2003, incorporated herein by reference in its entirety.

This application is related to PCT International Publication Numbers WO2005/020245 A2 and WO 2005/020245 A3, each of which is incorporatedherein by reference in its entirety

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not Applicable

INCORPORATION-BY-REFERENCE OF MATERIAL SUBMITTED ON A COMPACT DISC

Not Applicable

NOTICE OF MATERIAL SUBJECT TO COPYRIGHT PROTECTION

A portion of the material in this patent document is subject tocopyright protection under the copyright laws of the United States andof other countries. The owner of the copyright rights has no objectionto the facsimile reproduction by anyone of the patent document or thepatent disclosure, as it appears in the United States Patent andTrademark Office publicly available file or records, but otherwisereserves all copyright rights whatsoever. The copyright owner does nothereby waive any of its rights to have this patent document maintainedin secrecy, including without limitation its rights pursuant to 37C.F.R. § 1.14.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention pertains generally to superconducting power transmissioncables, and more particularly to conduction-cooling of high-temperaturesuperconducting power transmission cables.

2. Description of Related Art

Current high-temperature superconducting (HTS) power transmission cablesystems are cooled by convection using a coolant, such as liquidnitrogen, that circulates axially along the cable to keep the HTS cablebelow the superconducting transition temperature. This configurationrequires a cable construction resembling a hose that must contain thecoolant during operation. Coolant must be continually replenished insystems that do not recover the coolant. Cables are often hermeticallysealed and coolant recovered, so as to reduce the cost of coolantproduction. Current cable designs are often limited to lengths of lessthan 200 meters due to these limitations. There are two basic HTS cabledesigns recognized for electric power transmission: the Warm Dielectricdesign (WD) and the Cold Dielectric Coaxial design (CDC). Examples canbe found in Malozemoff AP, et al., “Power applications ofhigh-temperature superconductors: status and perspectives,” IEEE Trans.Appl. Supercond. 2002, 12(1): 778-781, and Kelley, N. et al.,“Application of HTS wire and cables to power transmission: state of theart and opportunities,” IIE Power Engineering Society Winter PowerMeeting, January 2001, which describe current designs forsuperconducting power transmission cables.

BRIEF SUMMARY OF THE INVENTION

The invention is principally related to axial cooling of asuperconducting cable. In general terms, the invention is aconduction-cooled superconducting power transmission cable wherein aHigh-Temperature Superconducting (HTS) wire is surrounded by an innerlayer of thermal insulator, one or more layers of highly thermalconducting material such as copper, and an outer layer of thermalinsulator with cryogenic coolant sources distributed along the powertransmission cable and coupled to the copper layers or both the copperlayers and the HTS wire. The copper layers remove ambient heat byconducting the heat axially along the cable thereby keeping the HTS wirebelow its superconducting transition temperature. The cryogenic coolantsources of the present invention can use helium, hydrogen, nitrogen,argon, neon, air, oxygen or mixtures thereof as a coolant. Cryogeniccoolant sources are placed along the cable at predetermined distancesfrom each other that may be greater than 50 meters, between a centimeterand 50 meters, or less than a centimeter. The cryogenic coolant sourcemay vary in scale from a macro scale (for power transmission greaterthan 50 meters) down to a nano scale (10-9 meters) for integration intomicroscopic instruments and equipment. The cryogenic coolant sources canbe coolant reservoirs, part of a coolant distribution system, or standalone or modular compressor systems, powered directly from electricitysourced from either the HTS wire or powered by external energy sources.Micro scale coolant sources can be Microelectromechanical systems (MEMS)and nano-scale coolant sources can be Nanoelectromechanical systems(NEMS).

Existing HTS power transmission cables use convection cooling with acryogen, typically liquid nitrogen, as a coolant that circulates withinthe cable and along the HTS wire. In contrast, the present inventionkeeps the HTS wire at temperatures below the critical transitiontemperature through the use of cryogenic coolant sources positioned atthe cable ends or distributed along the cable at periodic intervals, andcoupled to a highly conductive material, such as copper, radiallysurrounding the HTS wire. The heat that enters the cable radiallythrough the outer insulation layer is channeled axially, towards thecryogenic coolant source by the conducting copper layer instead ofallowing the heat to move in the radial direction towards the cold HTSwire. This configuration is feasible due to the high thermalconductivity of copper and copper alloys at cryogenic temperatures.

One distinct advantage of this invention is cryogenic coolant sourcesare positioned only at periodic locations along the HTS cable. There isno circulation of a cryogenic fluid coolant axially along the cable tokeep it cold and therefore no associated cost related to pumping andcontaining a cryogenic fluid in a cable.

One distinct advantage of this invention is cryogenic coolant sourcesare positioned only at periodic locations along the HTS cable. There isno circulation of a cryogenic fluid coolant axially along the cable tokeep it cold and therefore no associated cost related to pumping andcontaining a cryogenic fluid in a cable.

An aspect of the invention is a conduction-cooled superconducting powertransmission cable system that comprises a wire composed of ahigh-temperature superconducting material, an inner layer of thermalinsulation surrounding the wire, a layer of heat conducting materialsurrounding the inner layer of thermal insulation, an outer layer ofinsulation surrounding the layer of heat conducting material, and meansfor cooling the wire at or below its superconducting transitiontemperature, the means coupled to the layer of heat conducting material.

Another aspect of the invention is a cable system where thesuperconducting wire comprises a material having a transitiontemperature at or below approximately 110° K.

A further aspect is where the wire comprises a superconducting materialchosen from the group consisting of (Bi,Pb)₂Sr₂Ca₂Cu₃O₁₀, YBa₂Cu₃O₇ andMgB₂.

An aspect is where the heat conducting material comprises copper orcopper alloys.

Another aspect where the means for cooling comprises a cryogenic coolantsource.

A further aspect is where the cryogenic coolant source is adapted to usea coolant chosen from the group consisting of helium, hydrogen,nitrogen, argon, neon, air and oxygen.

A still further aspect is where the cryogenic coolant source is a standalone cryogenic compressor system.

Another aspect of the invention is where the cryogenic coolant source isa modular cryogenic refrigeration system.

Another aspect is where the cryogenic coolant source is adapted to coolthe cable system at length intervals of about 10 meters to at leastabout 200 meters.

A further aspect is where the cryogenic coolant source is adapted tocool the cable system at length intervals of about 1 meter to at leastabout 10 meters.

A still further aspect is where the cryogenic coolant source is adaptedto cool the cable system at length intervals of about less than 1 meter.

Another aspect of the invention is where the cryogenic coolant sourcefurther comprises a microelectromechanical system (MEMS).

A further aspect is where the cryogenic coolant source further comprisesa nanoelectromechanical system (NEMS).

Another aspect of the invention is where the cryogenic coolant source isadapted to use electricity sourced from the cable system or from anexternal source or both.

A further aspect is where the outer thermal insulation comprises one ormore layers of a highly reflective radiant barrier material alternatingwith one or more layers of a low-conductivity spacer material.

A still further aspect is where the highly reflective radiant barriermaterial comprises aluminum foil.

Another aspect is where the low conductivity spacer material comprisesfiberglass paper.

A further aspect is where the outer thermal insulation comprises athermal conductivity of about 3.7×10⁻⁵ W/m-K and a layer density ofabout 20 layer/cm.

A yet further aspect is where the radius of the wire is at least about0.1 centimeter to about 1 centimeter.

Another aspect of the invention is where the radius of the wire is lessthan or about 0.1 centimeter.

A further aspect is where the cross section of the wire is rectangular.

A still further aspect is where the outer radius of the inner thermalinsulation is at least about 2 centimeters.

Another aspect is where the outer radius of the outer thermal barrier isat least about 3 centimeters.

A further aspect is where the diameter of the cable system is at leastabout 20 centimeters.

A still further aspect is where the means for cooling is coupled to thewire.

Another aspect of the invention is a second layer of heat conductingmaterial surrounding the outer layer of thermal insulation, and a secondouter layer of insulation surrounding the second layer of heatconducting material.

An aspect of the invention is a conduction-cooled superconducting cablesystem, comprising a wire composed of a high-temperature superconductingmaterial, an inner layer of thermal insulation surrounding the wire, aplurality of heat conducting material layers surrounding the inner layerof thermal insulation, a plurality of insulation layers surrounding theplurality of heat conducting material layers, and means for cooling thewire at or below its superconducting transition temperature, the meansfor cooling coupled to the layers of heat conducting material atperiodic intervals along the superconducting cable system.

Another aspect of the invention is where the cryogenic coolant sourcesare adapted to cool the layers of heat conducting material at periodicintervals of at least about 1 meter and up to about 200 meters.

A further aspect is where the cryogenic coolant sources are adapted tocool the cable system at length intervals of about less than 1 meter.

Another aspect of the invention is where the cryogenic coolant sourcesfurther comprise a plurality of microelectromechanical systems (MEMS).

A further aspect is where the cryogenic coolant sources further comprisea plurality of nanoelectromechanical systems (NEMS).

An aspect of the invention is a method for fabricating aconduction-cooled high-temperature superconducting cable comprisingmanufacturing a high temperature superconducting wire, wrapping the wirewith an inner thermal barrier, encasing the inner thermal barrier with alayer of thermal conducting material, wrapping the layer of thermalconducting material with an outer thermal barrier, and coupling aplurality of cryogenic coolant sources to the layer of thermalconducting material at predetermined positions.

Another aspect of the invention is a method where the superconductingwire comprises a material having a transition temperature belowapproximately 110° K.

A further aspect is a method where the wire comprises a superconductingmaterial chosen from the group consisting essentially of(Bi,Pb)₂Sr₂Ca₂Cu₃O₁₀, YBa₂Cu₃O₇ and MgB₂.

A still further aspect is a method where the heat conducting materialcomprises copper or copper alloys.

Another aspect is a method where the outer thermal insulation compriseslayers of a highly reflective radiant barrier material alternating witha low-conductivity spacer material.

A further aspect is a method where the highly reflective radiant barriermaterial comprises aluminum foil and the low conductivity spacermaterial comprises fiberglass paper.

A yet further aspect is a method where the outer thermal insulationcomprises a thermal conductivity of about 3.7×10⁻⁵ W/m-K and a layerdensity of about 20 layer/cm.

Another aspect of the invention is a method where the cryogenic coolantsources are adapted to use a coolant chosen from the group consisting ofhelium, hydrogen, nitrogen, argon, neon, air and oxygen.

A further aspect is a method where the cryogenic coolant sources areadapted to cool the layers of heat conducting material at periodicintervals along the superconducting cable of at least about 1 meter andup to about 200 meters.

A still further aspect is a method where the cryogenic coolant sourcesare adapted to cool the cable system at periodic intervals along thesuperconducting cable of about less than 1 meter.

Another aspect is a method where the cryogenic coolant sources furthercomprise a plurality of microelectromechanical systems (MEMS).

A further aspect is a method where the cryogenic coolant sources furthercomprise a plurality of nanoelectromechanical systems (NEMS).

An aspect of the invention is a method for fabricating aconduction-cooled high-temperature superconducting cable that comprisesperforming heat transfer calculations to determine the dimensions of ahigh temperature superconducting wire, an inner thermal barrier, a layerof thermal conducting material, an outer thermal barrier, and spacingintervals of cryogenic coolant sources necessary to operate thesuperconducting cable below the critical temperature of the hightemperature superconducting wire, manufacturing the high temperaturesuperconducting wire, wrapping the wire with the inner thermal barrier,encasing the inner thermal barrier with the layer of thermal conductingmaterial, wrapping the layer of thermal conducting material with theouter thermal barrier, and coupling a plurality of the cryogenic coolantsources to the layer of thermal conducting material at calculatedspacing intervals.

Further aspects of the invention will be brought out in the followingportions of the specification, wherein the detailed description is forthe purpose of fully disclosing preferred embodiments of the inventionwithout placing limitations thereon.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)

The invention will be more fully understood by reference to thefollowing drawings which are for illustrative purposes only:

FIG. 1 is a perspective schematic view of a superconductor cable systemshowing the cable layers and cryogenic cooling sources at periodicintervals.

FIG. 2 is cross section schematic view of the superconductor cable shownin FIG. 1 and taken at line 2-2 showing the interface of the cryogeniccooling source and the heat conducting layer.

FIG. 3 is a cross section schematic view of the superconductor cableshown in FIG. 1 and taken at line 3-3 showing the position of themaximum temperature for the HTS wire.

FIG. 4 is a schematic cross section view of another embodiment of asuperconductor cable with the cryogenic cooling source connected only tothe heat conducting layer.

FIG. 5 is a schematic view of another embodiment of a superconductingcable with two heat conducting layers around the HTS wire.

DETAILED DESCRIPTION OF THE INVENTION

Referring more specifically to the drawings, for illustrative purposesthe present invention is embodied in the apparatus generally shown inFIG. 1 through FIG. 5. It will be appreciated that the apparatus mayvary as to configuration and as to details of the parts, and that themethod may vary as to the specific steps and sequence, without departingfrom the basic concepts as disclosed herein.

Several materials conduct electricity with zero resistance below acertain temperature, called critical transition temperature (T_(c)).Some examples of such superconducting materials are:(Bi,Pb)₂Sr₂Ca₂Cu₃O₁₀, also called BSCCO-2223 (T_(c)=110 K), YBa₂Cu₃O₇(T_(c)=90 K) and MgB₂ (T_(c)=39 K). Other superconducting materials witha T_(c) in the range of 39 K to 110 K may be used without departing fromthe teachings of the invention. Low cost cooling of superconductingdevices below the critical transition temperature is an important goalfor commercialization of superconducting power transmission systems.Existing superconductor systems require circulation of a coolant such asliquid helium (He) to keep the superconductor wire below the criticaltemperature. The emergence of high-temperature superconducting (HTS)material makes possible the use of liquid nitrogen,(N₂) hydrogen (H₂) orother materials with a relatively higher boiling point as a coolant insuperconducting transmission lines, thus reducing the cost of heatremoval with the coolant. Since in addition, HTS material has theadvantage of high power density and zero environmental impact, powertransmission lines using HTS cables are among the most promisingapplications of higher temperature superconductors.

In the present invention, a cable with multiple concentric layers has acore superconductor wire, one or more layers of copper, and multiplelayers of insulation arranged to keep the core superconductor wire belowits critical temperature. Cryogenic coolant sources are coupled to thecopper layers and/or the superconductor wire at periodic intervals alongthe cable and used to conduct heat away from the cable. Heat that entersthe cable radially through the outer insulation layer is channeledaxially towards the cable's cryogenic coolant sources by the copperlayers instead of allowing it to move radially inward towards the coldsuperconductor.

The thermal conductivity of copper and copper alloy varies withtemperature. In the cryogenic temperature range, the thermalconductivity of copper ranges from about 300 W/m-K at about 100 deg K toa high value of around 2000 W/m-K at about 20 deg K. This can be one tothree or more orders of magnitude greater than the thermal conductivityof HTS wire materials at their critical temperature.

A multilayer insulation consisting of alternating layers of a highlyreflecting material, such as aluminum foil, and a low-conductivityspacer, such as fiberglass paper is preferred. The value of the thermalconductivity of insulation used for the examples in this application andthe embodiments of this invention, k_(m)=3.7×10⁻⁵ W/m-K, corresponds toa multilayer insulation made with alternating layers of 0.006-mmaluminum foil and 0.15-mm fiberglass paper with a layer density of 20layer/cm.

FIG. 1 is a schematic perspective view of a High TemperatureSuperconducting (HTS) cable system generally designated 10 and inaccordance with the present invention. HTS cable body 12 comprises anouter insulation layer 14, a layer of heat conducting material 16positioned radially inward from outer insulation layer 14, an innerinsulation layer 18 positioned inward of heat conducting material 16 anda core of HTS wire 20. Although HTS wire 20 is illustrated here with acircular cross section, it can be other geometric cross sections such asrectangular, triangular or cylindrical without departing from theteachings of this invention. Positioned along the outside of cable body12 is a first cryogenic coolant source 22 having a first heat conduit 24that is coupled to heat conducting material 16 and HTS wire 20 (see FIG.2). First cryogenic coolant source 22 can be a reservoir of coolant,part of a distribution system of coolant, or a stand alone compressor orrefrigeration system for producing cryogenic coolant. First heat conduit24 is preferably a refrigerator system evaporator coil, but could alsobe a, a liquid coolant heat exchanger a convection heat pipe or athermal conductor between cryogenic coolant source 22 and heatconducting material 16. A second cryogenic coolant source 26 ispositioned on cable body 12 at a distance L from cryogenic coolantsource 22, and has a second heat conduit 28 that is also coupled to heatconduction material 16. Point M on cable body 12 is designatedequidistant between first and second heat conduit 24, 28. Point M is L/2distance from each heat conduit 24, 28.

Cryogenic coolant sources 22, 26 can use different coolants with boilingpoints in the cryogenic temperature range including helium, hydrogen,nitrogen, argon, neon, air, oxygen or mixtures thereof. Cryogeniccoolant sources 22, 26 can be mechanical or electromechanical and can beof a macro scale, micro scale or nano scale. Cryogenic coolant sources22, 26 are shown schematically positioned outside cable body 10 but maybe integrated into cable body 10 without departing from the teachings ofthe invention. Cryogenic coolant sources 22, 26 can be powered fromexternal sources of power or extract power from the HTS superconductorwire 20.

FIG. 2 is a schematic cross section view of HTS cable system 10 taken atline 2-2 in FIG. 1. Heat conduit 24 of first cryogenic coolant source 22penetrates outer insulation layer 14 and is in thermal contact with heatconducting material 16 at interface 30. Heat conduit 24 also penetratesinner insulation layer 18 and is in thermal contact with HTS wire 20 atinterface 32.

FIG. 3 is a schematic cross section view of HTS cable system 10 taken atline 3-3 in FIG. 1. Numerical analysis has determined that for thiscable configuration, the location of highest temperature on HTS wire 20will be midway between first, second cryogenic coolant sources 22, 26,designated as position M on the cable body 10 shown in FIG. 1, and atthe interface of HTS wire 20 and inner insulation 18. This positionwhere the highest or maximum temperature of the HTS wire 20 will occuris designated here as Tm. Superconducting cable configuration can beoptimized for a specific Tm with heat transfer calculations as will bediscussed below.

Referring back to FIG. 2, in a first embodiment of the invention, acable body 12 has an overall radius (R) of about 5 cm with a core HTSwire 20 of MgB₂ having a radius (r₁) of about 0.8 cm. The innerinsulation layer 18 is about 0.05 cm thick and a heat conducting copperlayer 16 is about 0.05 cm thick. First, second cryogenic coolant sources22, 26 use liquid He as the coolant and are positioned 12.5 metersapart, designated as length L in FIG. 1. For this configuration with anambient temperature of 300 deg K, the temperature at Tm will be about 15deg K, which is below the critical temperature for MgB₂ of 39 deg K.

For comparison, a superconducting cable of similar dimensions and lengthas above, but with the copper layer 16 replaced by an insulation layerwith thermal conductivity of k_(m)=3.7×10⁻⁵ W/m-K, will have atemperature Tm of over 100 deg K at Tm. The critical temperature forMgB₂ of 39 deg K. is exceeded and this configuration without the copperlayer will not support superconduction.

In a second embodiment of a superconducting cable 12 with about a 5 cmradius, the radius of the MgB₂ HTS wire 20 is 0.2 cm, the thickness ofthe inner insulation layer 18 is about 0.1 cm and the thickness of thecopper layer 16 is about 0.1 cm. With first, second cryogenic coolantsources 22, 24 using He coolant and spaced a distance L of about 12.5 mapart, Tm is about 7.4 deg K at an ambient temperature of 300 deg K.Thus the reduced radial dimensions of the HTS wire 20, inner insulationlayer 18 and copper layer 18 result in a lower core temperature for agiven cable length L.

In a third embodiment, the parameters of the second embodiment ofsuperconducting cable are the same except the length L between first,second cryogenic coolant sources 22, 26 has been increased to 50 meters.With ambient conditions of 300 deg K, the temperature at Tm at the HTSwire 20 and inner insulation 18 interface is about 28.2 deg K which isbelow the critical temperature for MgB₂ of 39 deg K. Thus a longerdistance between cryogenic coolant sources is facilitated by a smallerdiameter copper sheath and wire core. These three embodiments aresummarized below in Table 1. FIG. 4 is schematic cross section view ofanother mode of an HTS cable system 10 with a cryogenic coolant source40 having a heat conduit 42. Heat conduit 42 is configured to makethermal contact only with heat conducting material 16 at interfaceposition 44.

In further embodiments according to the invention, BSSCCO-2233, (T_(c)110 deg K), is used as HTS core wire 20. The relatively higher operatingtemperature of this superconductor wire allows a cable system to beoperated with cryogenic coolant sources using Hydrogen (H₂), (boilingpoint 20.3 deg K), Nitrogen (N₂) (boiling point 77.4 deg K) or othergases or gas mixtures with a boiling point below the criticaltemperature of BSSCCO-2233 of 110 deg. K.

Using higher operating temperatures for superconducting systems providessignificant savings in coolant operating costs. For example, the Carnotcoefficient of performance for He is 5.1 times lower than the one for H₂which means increased refrigeration work is required with He as acoolant. The Carnot coefficient of performance for H₂ is 4.8 times lowerthan the one for N₂. In other words, it takes about 25 times the energyto remove one watt of heat at 4.2 deg K with He as a coolant compared toremoving one watt of heat at 77 deg K with N₂. From the viewpoint ofrefrigeration work required to keep the temperature of thesuperconductor wire below its critical temperature, nitrogen ispreferable to hydrogen, and hydrogen is preferable to helium. On theother hand, operation using nitrogen as a coolant requires installationof a higher number of refrigeration systems or cryogenic coolantsources, located at shorter periodic length intervals, than usinghydrogen and helium; however each of these nitrogen sources has asmaller power requirement.

In a fourth embodiment of the invention, a cable body 12 with a radiusof about 10 cm has a core of HTS wire 20 of BSSCCO-2233 with a radius ofabout 0.1 cm, an inner insulation layer 18 with a thickness of about 0.1cm, a copper layer 16 with a thickness of about 0.2 cm and an outerinsulation layer 14. First, second cryogenic coolant sources 22, 24employ liquid H₂ as a coolant and are positioned at distance L about 200meters apart. For this configuration, Tm will be about 87.5 deg K whenthe ambient temperature is about 300 deg K. In a fifth embodiment, thecopper layer 16 is reduced to a thickness of about 0.1 cm, and theremaining parameters remain the same. Tm will be about 157 deg K, whichis above the critical temperature required by HTS wire 20. Thus thisfifth embodiment will not support superconduction.

However, in a sixth embodiment, if the copper layer 16 is increased to athickness of about 0.4 cm and the remainder of cross section dimensionsof cable components remain the same as above, Tm will be about 63 deg Kwhich is below the critical temperature of 110 deg K.

In order to compare the efficacy of different coolants, Table 2summarizes the cable system embodiments described above with differentcopper layer thicknesses. Embodiments 4 through 6 use He as a coolant.Embodiments 7 through 9 use H₂ as a coolant. The heat load for the twocryogenic coolant sources combined (one at each cable segment end) isexpressed in watts (W).

As can be seen, for embodiments 4 and 7 where the thickness of thecopper layer is 0.1 cm, Tm is shown with an * and is above the criticaltemperature required. These embodiments will not supportsuperconduction. For embodiments 5 and 8, the thickness of the copperlayer is about 0.2 cm and H₂ coolant uses less energy than He. Forembodiments 6 and 9, the copper layer is about 0.3 cm and a lowertemperature at Tm is maintained compared with embodiments 5 and 8respectively. H₂ coolant using less energy than He in embodiment 9compared to embodiment 6.

Further embodiments of the invention use N₂ as a coolant. Referring toFIG. 1, embodiment 10 through 12 have a cable body 12 with a radius (R)of about 10 cm, a core of HTS wire 20 of BSSCCO-2233 with a radius (r₁)of about 0.1 cm, an inner insulation layer 18 with a thickness of about0.1 cm, a copper layer 16 and an outer insulation layer 14. Cryogeniccoolant sources 22, 26 employ N₂ and here are spaced a distance L about50 m apart. Table 3 summarizes the maximum temperature Tm for the HTSwire and heat load on the combined cryogenic coolant sources using N₂ asa coolant with different thicknesses of copper layer 16.

As can be seen in Table 3, a superconducting cable using cryogeniccoolant sources employing N₂ and spaced 50 meters apart can maintain Tmbelow the critical temperature required. It can be shown that for anequivalent 200 meter cable, four, 50 meter segments with N₂ cryogeniccoolant sources (Table 3) will use less energy than one 200 metersegment with He or H₂ cryogenic coolant sources at each end (Table 2).

FIG. 5 illustrates a cross section view of another mode of the inventionemploying multiple layers of copper in the cable. Proceeding radiallyinward, cable body 100 has an outer insulation layer 110, a firstconducting layer 112, an interstitial insulation layer 114, a secondconducting layer 116, an inner insulation layer 118 and an HTS core wire120. In one aspect of this mode (not shown), HTS wire 120, and first andsecond conducting layers 112, 116 are coupled to cryogenic coolantsources in a manner previously described in FIG. 2. In another aspect ofthis mode (not shown), only first and second copper layers 112, 116 arecoupled to the cryogenic coolant sources. Multiple conducting layers canhave an affect on the time required to cool a superconductor cable downfrom ambient temperature as will be discussed below.

The configuration of a superconducting cable of the present inventioncan be optimized using heat transfer models of axial conduction coolingas described below.

Equation (1) gives the thermal conductivity k_(c) (W/m-K) for an averagesample of oxygen-free copper, as a function of temperature, T (K).$\begin{matrix}{{\log\quad k_{c\quad}} = \frac{\begin{matrix}{2.2154 - {0.88068 \cdot T^{0.5}} + {0.29505 \cdot T} -} \\{{0.048310 \cdot T^{\quad 1.5}} + {0.003207 \cdot T^{2}}}\end{matrix}}{\begin{matrix}{1 - {0.47461 \cdot T^{0.5}} + {0.13871 \cdot T} -} \\{{0.020430 \cdot T^{\quad 1.5}} + {0.001281 \cdot T^{2}}}\end{matrix}}} & (1)\end{matrix}$

The thermal conductivity for this material can vary widely dependingupon the residual resistivity ratio, RRR. The values calculated withequation (1), are for RRR=100. It should be noted that the thermalconductivity of copper at cryogenic temperatures can reaches values over2000 W/m-K at about 20 deg K.

A multilayer insulation consisting of alternating layers of a highlyreflecting material, such as aluminum foil, and a low-conductivityspacer, such as fiberglass paper is used. The value of the thermalconductivity used for this component is, k_(m)=3.7×10⁻⁵ W/m-K, andcorresponds to a multilayer insulation made with the materials mentionedabove, with a layer density of 20 layer/cm.

HTS wire MgB₂ has a critical temperature of 39 K. Its thermalconductivity, in k_(s) (W/m-K), is approximated by equation (2a) where Tis in K.log k _(s)=−1.6158−3.8472·log T+18.003·(log T)²−21.307−(logT)³+12.111·(log T)⁴−3.4362·(log T)⁵+0.3905·(log T)⁶   (2a)

The HTS material BSCCO-2223 ((Bi,Pb)₂Sr₂Ca₂Cu₃O₁₀,), has a criticaltemperature T_(c)=110 deg K. Its thermal conductivity, k_(s) (W/m-K), inthe range from 20 K to 220 K is represented by equation (2b), where T isin K.log k _(s)=−651.61+2308.7·log T−3371.8·(log T)²+2598.2·(logT)³−1113.8·(log T)⁴+251.85·(log T)⁵−23.475·(log T)⁶   (2b)

At 90 deg K, the thermal conductivity of BSCCO is 4.2 W/m-K where thecorresponding thermal conductivity of copper is 487 W/m-K (more than 100times larger). At 30 deg K, the thermal conductivity of BSCCO is 2.9W/m-K where the corresponding thermal conductivity of copper is 2143W/m-K (more than 700 times larger).

Considering steady state wire operation, as well as no angulardependence of the boundary conditions and of the thermal properties ofall employed materials, heat transfer behavior, in each of the cable'sregions, is captured by the following differential equation:$\begin{matrix}{0 = {{\frac{1}{r}\frac{\partial\left( {r \cdot q_{r}} \right)}{\partial r}} + \frac{\partial\left( q_{z} \right)}{\partial z}}} & (3)\end{matrix}$where q_(r) and q_(z) represent heat fluxes in the directions r and z,respectively. In turn, q_(r) and q_(z) are given by the expressions:$\begin{matrix}{q_{r} = {{- k}\quad\frac{\partial T}{\partial r}}} & (4) \\{q_{z} = {{- k}\frac{\partial T}{\partial z}}} & (5)\end{matrix}$where k represents the thermal conductivity of the material and T istemperature. After substitution of (4) and (5), equation (3) can berewritten as: $\begin{matrix}{0 = {{\frac{\partial}{\partial r}\left( {r \cdot k \cdot \frac{\partial T}{\partial r}} \right)} + {\frac{\partial}{\partial z}\left( {r \cdot k \cdot \frac{\partial T}{\partial z}} \right)}}} & (6)\end{matrix}$

We consider the following boundary conditions:T(z=±L/2)=T _(o) ∀rε[0,R]  (7)T(r=R)=T _(e) ∀zε[−L/2, L/2]  (8)where T_(e) is the ambient temperature (T_(e)=300 K), and T_(o) is thelow temperature imposed at the extremes of the cable (i.e T_(o)=4.2 Kfor He).

Due to the symmetry at z=0 (position M in FIG. 1), the calculationdomain work is reduced by considering only the solution for z ε [0,L/2], and a new boundary condition: $\begin{matrix}\begin{matrix}{{\frac{\partial T}{\partial z}\left( {z = 0} \right)} = 0} & \quad & {\forall{r \in \left\lbrack {0,R} \right\rbrack}}\end{matrix} & (9)\end{matrix}$

Introducing the dimensionless variables: $\begin{matrix}{r^{*} = \frac{r}{R}} & \quad & {z^{*} = {2 \cdot \frac{z}{L}}} & \quad & {T^{*} = \frac{T - T_{o}}{T_{e} - T_{o}}}\end{matrix}$equation (4) can be written as: $\begin{matrix}{0 = {{\frac{\partial}{\partial r^{*}}\left( {r^{*} \cdot \frac{k}{R^{2}} \cdot \frac{\partial T^{*}}{\partial r^{*}}} \right)} + {\frac{\partial\quad}{\partial z^{*}}\left( {r^{*} \cdot \frac{4 \cdot k}{L^{2}} \cdot \frac{\partial T^{*}}{\partial z^{*}}} \right)}}} & (10)\end{matrix}$

This equation is valid in each material as long as the proper value ofthermal conductivity is used. Therefore we can write equation (10) forfour different subdomains (i=1, . . . 4), obtaining a system of fourdifferential equations, as indicated below: $\begin{matrix}{{0 = {{\frac{\partial\quad}{\partial r^{*}}\left( {r^{*} \cdot \frac{k_{i}}{R^{2}} \cdot \frac{\partial T_{i}^{*}}{\partial r^{*}}} \right)} + {\frac{\partial\quad}{\partial z^{*}}\left( {r^{*} \cdot \frac{4 \cdot k_{i}}{L^{2}} \cdot \frac{\partial T_{i}^{*}}{\partial z^{*}}} \right)}}}{{{{for}\quad i} = 1},{\ldots\quad 4.}}} & \left( {10\text{-}i} \right)\end{matrix}$where T_(i)* is the dimensionless temperature in subdomain i and k_(i)is the thermal conductivity in subdomain i (k_(i)=k_(i)(T_(i)*)). Thusequation (10-i) is valid for subdomain i. subdomain 1 ([0, r₁*=r₁/R]) isthe superconductor wire; subdomain 2 ([r₁*, r₂*=r₂/R]) is the innerinsulation layer; subdomain 3 ([r₂*, r₃*=r₃/R]) is the copper layer; andsubdomain 4 ([r₃*, 1]) is the outer insulation layer. k₁; k₂ and k₄; k₃are the thermal conductivities of superconductor; thermal insulator;copper respectively.

The resulting system of four second-order differential equations issolved for T₁*, T₂*, T₃* and T₄*, as functions of r* and z*, whencombined with four boundary conditions in each subdomain (16 in total).In addition to the boundary conditions implied by (7) and (8), and thosedue to the symmetry of the cylinder ((12) and (16-i)), we consider thecontinuity of the temperature ((13-i)) and the continuity of the radialheat flux ((14-i)) (shown below) at the interface between twosubdomains. The complete set of 16 boundary conditions is:$\begin{matrix}\begin{matrix}{{T_{4}^{*}\left( {r^{*} = 1} \right)} = 1} & \quad & {\forall{z^{*} \in \left\lbrack {0,1} \right\rbrack}}\end{matrix} & (11) \\\begin{matrix}{{\frac{\partial T_{1}^{*}}{\partial r^{*}}\left( {r^{*} = 0} \right)} = 0} & \quad & {\forall{z^{*} \in \left\lbrack {0,1} \right\rbrack}}\end{matrix} & (12) \\{{T_{i}^{*}\left( {r^{*} = r_{i}^{*}} \right)} = {T_{i + 1}^{*}\left( {r^{*} = r_{i}^{*}} \right)}} & \left( {13\text{-}i} \right) \\\begin{matrix}{\forall\quad{z^{*}\quad \in \quad\left\lbrack {0,\quad 1} \right\rbrack}} & \quad & {{{{for}\quad i}\quad = \quad 1},\quad{\ldots\quad 3}}\end{matrix} & \quad \\{{k_{i}\frac{\partial T_{i}^{*}}{\partial r^{*}}\left( {r^{*} = r_{i}^{*}} \right)} = {k_{i + 1}\frac{\partial T_{i + 1}^{*}}{\partial r^{*}}\left( {r^{*} = r_{i}^{*}} \right)}} & \left( {14\text{-}i} \right) \\\begin{matrix}{\forall{z^{*} \in \left\lbrack {0,1} \right\rbrack}} & \quad & {{{{for}\quad i} = 1},{\ldots\quad 3}}\end{matrix} & \quad \\\begin{matrix}{{T_{i}^{*}\left( {z^{*} = 1} \right)} = 0} & \quad & {\forall{r^{*} \in {{Subdomain}\quad i}}} & \quad & {{{{for}\quad i} = 1},{\ldots\quad 4}}\end{matrix} & \left( {15\text{-}i} \right) \\\begin{matrix}{{\frac{\partial T_{i}^{*}}{\partial z^{*}}\left( {z^{*} = 0} \right)} = 0} & \quad & {\forall{r^{*} \in {{Subdomain}\quad i}}} & \quad & {{{{for}\quad i} = 1},{\ldots\quad 4}}\end{matrix} & \left( {16\text{-}i} \right)\end{matrix}$

For proper operation of a superconducting cable, it is necessary tomaintain the temperature of the superconducting wire below its criticaltransition temperature. The temperature profile in the superconductingwire can be found by solving the system of equations (10-i)(i=1, . . .4) for a particular cable configuration (specific values of r_(i)* andL/R). In the embodiments of the invention with a copper layer, themaximum temperature in the temperature profile along the wire, Tm, isdetermined to occur at the midpoint between the cryogenic coolantsources and at the interface of the superconductor wire and the innerinsulation as described previously. Different cable configurations canbe evaluated and those that result in maximum temperatures Tm below thesuperconductor's transition temperature can be used for superconductingpower transmission.

The program FEMLAB® version 2.3 may be used to perform the numericalcalculations above. With this software, the aforementioned PDE (PartialDifferential Equation) problem is approximated using the Finite ElementMethod (FEM). FEMLAB® generates automatically a triangular mesh thatcovers the domain under consideration and takes into account the problemgeometry. The solution is represented as a weighted linear combinationof a linearly independent set of piecewise linear polynomials φ_(k)(z*,r*). This approximation requires that the polynomials be pieced togetherin such a manner that the resulting function is continuous with anintegrable or continuous first or second derivative on the entiredomain, and takes the form: $\begin{matrix}{{T^{*}\left( {z^{*},r^{*}} \right)} = {\sum\limits_{k = 1}^{m}{U_{k} \cdot {\varphi_{k}\left( {z^{*},r^{*}} \right)}}}} & (17)\end{matrix}$where m is the total number of elements in the mesh (triangles) andU_(k) are coefficients that are determined so as to vanish theprojection of the approximation error on the finite dimensional spacespanned by the approximating polynomials φ_(k).

The approximate solution can be improved by increasing the number ofelements in the mesh (refining the mesh), i.e. by dividing the elementsinto smaller elements. Such a refined mesh requires longer calculationtime and higher computer memory availability. This simulation tool canbe used to confirm that the highest temperature in the HTSsuperconductor wire is attained at the interface with the innerinsulation layer and in the middle of the axial length of the wire thefurthest location from the cryogenic coolant sources (Tm in FIG. 3).

This simulation tool FEMLAB® can also be used to verify that inembodiment 1, if copper is not used to channel the radial heat axially,but additional thermal insulator is used instead of a copper layer,i.e.: k₂=k₃=k₄=k_(m)=3.7×10⁻⁵ W/m-K, in the equations above, the highesttemperature in the HTS superconductor wire using He as a coolant will beapproximately 104 deg K, and above the critical superconductingtemperature of 39 deg K for MgB₂.

The simulation tool can also be used to show that the axial heat flux ishigher in the copper layer than in the other materials (close to 3orders of magnitude higher when compared with the superconductor and 7orders with respect to the thermal insulation layers), as expected dueto copper's thermal conductivity. The copper layer is the preferredtrack for the heat, in such a way that most of the radial heat thatpasses through the outer insulation layer, coming from the environment,finds an easier way to continue along the copper towards the coldextremes, making this heat flux bigger, in general, as we move in thisdirection. Integration of the axial heat flux over the boundary of thecable at the cold extreme gives an approximation to the total heat flowtransferred to the liquid helium at one extreme. For the firstembodiment above using MgB₂ as the HTS wire and He as the coolant, thetotal heat flow is calculated to be about 3.5 W, and over 95% of thisheat is calculated to come from the copper layer.

Applying the equations above to the first embodiment, an insulated cablewith radius, R, of 5 cm, the length of the cable L=12.5 m, and theradius of a superconductor wire of MgB₂, r₁=4 cm, can be kept below 14.5K, which is a temperature low enough to assure the superconducting stateof the wire.

An insulated cable with the same values for R and L, as above can bekept at even lower temperature if the radius of the superconductor wireis decreased and the thickness of the outer insulation layer isincreased. For example, embodiment 2 with an insulated cable withradius, R, of 5 cm, the length of the cable L=12.5 m, and the radius ofa superconductor wire of MgB₂, r₁=2 cm, yields a calculated maximumtemperature of about 7.2 K for the wire.

In an example of optimizing cable length, the third embodiment discussedabove has an insulated cable with radius R=10 cm, length L=100 m, andradius of the superconductor wire r₁=2 cm, and yields a calculatedmaximum temperature below about 25.9 K which is a temperature lower thanT_(c) for the employed HTS material MgB₂. The heat transfer model andsimulation tools may also be used to calculate the operating conditionswhen using H₂ and N₂ as coolants.

Before a superconductor cable can be energized, the HTS wire must becooled from an ambient state to below the critical temperature.Numerical analysis has determined that the time required to cool asuperconducting cable from ambient condition to below criticaltemperature is reduced by using multiple layers of copper in the cableand coupled to the cryogenic coolant sources.

The time period of a cable's initial cool down from ambient temperatureto operating temperature may be determined by carrying out a detailedcalculation of heat transfer in the cable under transient conditions.These calculations determine the time required to cool down thesuperconducting wire to a temperature below it's critical value.BSCCO-2223 is used in the calculation examples below since long lengthBSCCO-2223 wire is readily available and manufactured by a number ofcompanies worldwide.

First the properties of the materials are determined. The values forthermal conductivity of copper, and the BSCCO-2223 wire are expressed inequations (1) and (2b) above. A multilayer insulation consisting ofalternating layers of 0.006-mm aluminum foil and 0.15-mm fiberglasspaper has a thermal conductivity of, k_(m)=3.7×10⁻⁵ W/m-K, andcorresponds to a multilayer insulation made with a layer density of 20layer/cm.

Equation (18) gives a correlation for the specific heat of copper,cp_(c) (J/kg-K), as a function of temperature, T (K).log c_(p) _(m) =−1.91844−0.15973·log T+8.61013·(log T)²−18.99640·(logT)³+21.96610·(log T)⁴−12.73280·(log T)⁵+3.54322·(log T)⁶−0.37970·(logT)⁷   (18)

The mass density of copper, p_(c), is approximated to 8933 kg/m³.

The specific heat of the multilayer insulation, cp_(m), is approximatedfrom each component's specific heat and mass fraction as follows:c _(p) _(m) =mf _(A1) ·cp _(A1)+(1−mf _(A1))·c _(p) _(fg)   (19)

The specific heats of aluminum, cp_(A1) (J/kg-K), and fiberglass,cp_(fg)(J/kg-K), are correlated with temperature, T (K):log c_(p) _(A1) =46.6467−314.292·log T+866.662·(log T)²−1298.30·(logT)³+1162.27·(log T)⁴−637.795(log T)⁵+210.351·(log T)⁶−38.3094·(logT)⁷+2.96344·(log T)⁸   (20)log c _(p) _(fg) =−2.4083+7.6006·log T−8.2982·(log T)²+7.3301·(logT)³−4.2386·(log T)⁴+1.4294·(log T)⁵−0.24396·(log T)⁶+0.015236·(log T)⁷  (21)

Using mass densities of 2702 kg/m³ for aluminum and 32 kg/m³ forfiberglass, the mass fraction of aluminum foil in the composite materialis mf_(A1)=0.7716, and the bulk mass density of the multilayerinsulation is calculated as p_(m)=42.024 kg/m³.

The specific heat, c_(p) _(s) (J/mol-K), for BSCCO-2223 in the rangefrom 20 K to 300 K is represented by equation, (22) where T is in K.log c_(p) _(s) =−6.6155+9.3975·log T−3.2259·(log T)²+0.3796·(log T)³  (22)

Calculation of specific heat per mass of superconductor is obtainedusing an approximate molecular weight for BSCCO-2223 of 1023.28 g/mol.The mass density of the superconductor is approximately p_(s)=6310kg/m³.

For time-dependent wire operation, with no angular dependence of theboundary conditions and of the thermal properties of all employedmaterials, heat transfer behavior, in each of the cable's regions, iscaptured by the differential equation: $\begin{matrix}{{\rho \cdot c_{p} \cdot \frac{\partial T}{\partial t}} = {- \left\lbrack {{\frac{1}{r}\frac{\partial\left( {r \cdot q_{r}} \right)}{\partial r}} + \frac{\partial\left( q_{z} \right)}{\partial z}} \right\rbrack}} & (23)\end{matrix}$where p is the mass density of the material, c_(p) represents itsspecific heat, T(t,r,z) is the temperature as a function of time t,radial distance r, and axial distance z. q_(r) and q_(z) represent heatfluxes in the directions r and z, respectively.

In turn, q_(r) and q_(z) are given by the expressions: $\begin{matrix}{q_{r} = {{- k}\frac{\partial T}{\partial r}}} & (24) \\{q_{z} = {{- k}\frac{\partial T}{\partial r}}} & (25)\end{matrix}$where k represents the thermal conductivity of the material. Aftersubstitution of (24) and (25), equation (23) can be rewritten as:$\begin{matrix}{{\rho \cdot c_{p} \cdot r \cdot \frac{\partial T}{\partial t}} = {{\frac{\partial}{\partial r}\left( {r \cdot k \cdot \frac{\partial T}{\partial r}} \right)} + {\frac{\partial}{\partial z}\left( {r \cdot k \cdot \frac{\partial T}{\partial z}} \right)}}} & (26)\end{matrix}$

The following are initial and boundary conditions: $\begin{matrix}\begin{matrix}{{T\left( {0,r,z} \right)} = T_{e}} & {{\forall{r \in \left\lbrack {0,R} \right\rbrack}};{\forall{z \in \left\lbrack {{{- L}/2},{L/2}} \right\rbrack}}}\end{matrix} & (27) \\\begin{matrix}{{T\left( {t,r,{{\pm L}/2}} \right)} = T_{o}} & {{\forall{r \in \left\lbrack {0,R} \right\rbrack}};{\forall{t \in \left\lbrack {0,\infty} \right)}}}\end{matrix} & (28) \\\begin{matrix}{{T\left( {t,R,z} \right)} = T_{e}} & {{\forall{z \in \left\lbrack {{{- L}/2},{L/2}} \right\rbrack}};{\forall{t \in \left\lbrack {0,\infty} \right)}}}\end{matrix} & (29) \\\begin{matrix}{{\frac{\partial T}{\partial r}\left( {t,0,z} \right)} = 0} & {{\forall{z \in \left\lbrack {{{- L}/2},{L/2}} \right\rbrack}};{\forall{t \in \left\lbrack {0,\infty} \right)}}}\end{matrix} & (30)\end{matrix}$where T_(e) is the ambient temperature (T_(e)=300 K), and T_(o) is thelow temperature imposed at the extremes of the cable (T_(o)=77.4 K ifliquid nitrogen is used or T_(o)=20.3 K if liquid hydrogen is used).

Due to the symmetry at z=0, the point midway between the cryogeniccoolant sources, we can reduce the calculation domain by consideringonly the solution for z ε [0, L/2], and a new boundary condition:$\begin{matrix}\begin{matrix}{{\frac{\partial T}{\partial z}\left( {t,r,0} \right)} = 0} & {{\forall{r \in \left\lbrack {0,R} \right\rbrack}};{\forall{t \in \left\lbrack {0,\infty} \right)}}}\end{matrix} & (31)\end{matrix}$

Introducing the dimensionless variables: $r^{*} = \frac{r}{R}$$z^{*} = {2 \cdot \frac{z}{L}}$$T^{*} = \frac{T - T_{o}}{T_{e} - T_{o}}$ $t^{*} = \frac{t}{t_{f}}$where t_(f) is final time, equation (26) can be written as:$\begin{matrix}{{\frac{\rho \cdot c_{p} \cdot L^{2}}{t_{f}} \cdot r^{*} \cdot \frac{\partial T^{*}}{\partial t^{*}}} = {{\frac{\partial}{\partial r^{*}}\left( {\left( \frac{L}{R} \right)^{2} \cdot k \cdot r^{*} \cdot \frac{\partial T^{*}}{\partial r^{*}}} \right)} + {\frac{\partial}{\partial z^{*}}\left( {4 \cdot k \cdot r^{*} \cdot \frac{\partial T^{*}}{\partial z^{*}}} \right)}}} & (32)\end{matrix}$

This equation is valid in each material as long as the proper values ofmass density, specific heat and thermal conductivity are used. Thereforewe can write equation (32) for five different subdomains (i=1, . . . 5),obtaining a system of five differential equations, as indicated below:$\begin{matrix}{{{\frac{\rho_{i} \cdot c_{p_{i}} \cdot L^{2}}{t_{f}} \cdot r^{*} \cdot \frac{\partial T_{i}^{*}}{\partial t^{*}}} = {{{\frac{\partial}{\partial r^{*}}\left( {\left( \frac{L}{R} \right)^{2} \cdot k_{i} \cdot r^{*} \cdot \frac{\partial T_{i}^{*}}{\partial r^{*}}} \right)} + {\frac{\partial}{\partial z^{*}}\left( {4 \cdot k_{i} \cdot r^{*} \cdot \frac{\partial T_{i}^{*}}{\partial z^{*}}} \right)\quad{for}\quad i}} = \quad 1}},{\ldots\quad 5}} & {(33)\text{-}i}\end{matrix}$where T_(i)* is the dimensionless temperature in subdomain i, and p_(i),c_(pi) and k_(i) are mass density, specific heat and thermalconductivity, respectively, in subdomain i (c_(pi=c) _(pi)(T_(i)*),k_(i)=k_(i)(T_(i)*)). Thus equation (33)-i is valid for subdomain i.Subdomain 1 ([0, r₁*=r₁/R]) is the superconductor wire; subdomain 2([r₁*, r₂=r₂/R]) is the inner copper layer; subdomain 3 ([r₂*,r₃*=r₃/R]) is the inner insulation layer; subdomain 4 ([r₃*, r₄*=r₄/R])is the outer copper layer; and subdomain 5 ([r₄*, 1]) is the outerinsulation layer. k₁l; k₂ and k₄; k₃ and k₅ are the thermalconductivities of superconductor; copper; thermal insulatorrespectively, and the same correlation applies to densities and specificheats.

The resulting system of five second-order differential equations issolved for T₁*, T₂*, T₃*, T₄* and T₅* as functions of r*, z* and t*,when combined with four boundary conditions and one initial condition ineach subdomain (a total of 20 boundary conditions and 5 initialconditions). In addition to the boundary conditions implied by (28) to(31), we consider the continuity of the temperature (36)-i) and thecontinuity of the radial heat flux ((37)-i) (described below) at theinterface between two subdomains. Therefore the complete set of boundaryand initial conditions is: $\begin{matrix}\begin{matrix}{{T_{5}^{*}\left( {t^{*},1,z^{*}} \right)} = 1} & {{\forall{z^{*} \in \left\lbrack {0,1} \right\rbrack}};{\forall{t^{*} \in \left\lbrack {0,1} \right\rbrack}}}\end{matrix} & (34) \\\begin{matrix}{{\frac{\partial T_{1}^{*}}{\partial r^{*}}\left( {t^{*},0,z^{*}} \right)} = 0} & {{\forall{z^{*} \in \left\lbrack {0,1} \right\rbrack}};{\forall{t^{*} \in \left\lbrack {0,1} \right\rbrack}}}\end{matrix} & (35) \\{{\begin{matrix}{{T_{i}^{*}\left( {t^{*},r_{i}^{*},z^{*}} \right)} = {T_{i + 1}^{*}\left( {t^{*},r_{i}^{*},z^{*}} \right)}} & {{\forall{z^{*} \in \left\lbrack {0,1} \right\rbrack}};{\forall{t^{*} \in \left\lbrack {0,1} \right\rbrack}}} & {{{for}\quad i} =}\end{matrix}1},{\ldots\quad 4}} & {(36)\text{-i}} \\\begin{matrix}{{k_{i}\frac{\partial T_{i}^{*}}{\partial r^{*}}\left( {t^{*},r_{i}^{*},z^{*}} \right)} = {k_{i + 1}\frac{\partial T_{i + 1}^{*}}{\partial r^{*}}\left( {t^{*},r_{i}^{*},z^{*}} \right)}} & {{\forall{z^{*} \in \left\lbrack {0,1} \right\rbrack}};{\forall{t^{*} \in \left\lbrack {0,1} \right\rbrack}}} & {{{{for}\quad i} =}{1,{\ldots\quad 4}}}\end{matrix} & {(37)\text{-}\text{i}} \\\begin{matrix}{{T_{i}^{*}\left( {t^{*},r^{*},1} \right)} = 0} & {{\forall{r^{*} \in {{Subdomain}\quad i}}};{\forall{t^{*} \in \left\lbrack {0,1} \right\rbrack}}} & {{{{for}\quad i} = 1}{,{\ldots\quad 5}}}\end{matrix} & {(38)\text{-i}} \\\begin{matrix}{{\frac{\partial T_{i}^{*}}{\partial z^{*}}\left( {t^{*},r^{*},0} \right)} = 0} & {{\forall{r^{*} \in {{Subdomain}\quad i}}};{\forall{t^{*} \in \left\lbrack {0,1} \right\rbrack}}} & {{{{{for}\quad i} = 1},}{\ldots\quad 5}}\end{matrix} & {(39)\text{-i}} \\\begin{matrix}{{T_{i}^{*}\left( {0,r^{*},z^{*}} \right)} = 1} & {{\forall{r^{*} \in {{Subdomain}\quad i}}};{\forall{z^{*} \in \left\lbrack {0,1} \right\rbrack}}} & {{{{for}\quad i} = 1}{,{\ldots\quad 5}}}\end{matrix} & {(40)\text{-i}}\end{matrix}$

In order to attain a superconducting state in the cable, it is necessaryto cool down the superconducting wire from the initial ambienttemperature (T_(e)=300 K) to a temperature below its criticaltemperature. The temperature profile in the superconducting wire at anytime during the cooling process can be found by solving the system ofequations (33)-i (i=1, . . . 5) for a particular cable configuration(specific values of r_(i)*, L/R and L).

Referring back to FIG. 5, embodiment thirteen is a cable body 100 with a10 cm radius (R) and comprises an outer insulation layer 110, a firstcopper layer 112 with a thickness of about 0.05 cm, an interstitialinsulation layer 114 with a thickness of about 0.05 cm, a second copperlayer 116 of about 0.05 cm, an inner insulation layer 118 with athickness of about 0.05 cm and an HTS wire 120 of BSSCCO-2233 with aradius (r₁) of about 0.1 cm. Cryogenic coolant sources employing H₂ arecoupled to the first, second copper layer 112, 116 and to HTS wire 120and spaced distance L about 10 meters apart on the cable in a mannerpreviously described in FIG. 1. With this configuration, it will takeapproximately 26 hours to cool the core of HTS wire 120 at a pointmidway between the cryogenic coolant sources to about to about 96.6 degK. In a fourteenth embodiment, cryogenic coolant sources employ N₂, andwith the same cable dimensions as above, take about 72 hours to cool thecore of HTS wire 120 to about 97.2 deg K. In a fifteenth embodiment,cryogenic coolant sources employing H₂ with the same cable dimensions asbefore but spaced distance L about 150 meters apart, take approximately58 hours to cool the HTS wire 120 to about 97.2 deg K.

For comparison, it is estimated that a superconductor cable configuredas embodiment 13 but with 1 layer of copper may take as long as 24 tocool down to critical temperature.

Table 4 summarizes the calculations of the cool down period for thethree aforementioned embodiments of a superconducting cable using twolayers of copper described above.

Other combinations of multiple copper layers and intervals of cryogeniccoolant sources can be specified and calculations performed to balancefirst cost of the cable system, operation cost of the cryogenic coolantsources and time to cool to critical temperature without departing fromthe teachings of this invention.

Although the description above contains many details, these should notbe construed as limiting the scope of the invention but as merelyproviding illustrations of some of the presently preferred embodimentsof this invention. Therefore, it will be appreciated that the scope ofthe present invention fully encompasses other embodiments which maybecome obvious to those skilled in the art, and that the scope of thepresent invention is accordingly to be limited by nothing other than theappended claims, in which reference to an element in the singular is notintended to mean “one and only one” unless explicitly so stated, butrather “one or more.” All structural, chemical, and functionalequivalents to the elements of the above-described preferred embodimentthat are known to those of ordinary skill in the art are expresslyincorporated herein by reference and are intended to be encompassed bythe present claims. Moreover, it is not necessary for a device or methodto address each and every problem sought to be solved by the presentinvention, for it to be encompassed by the present claims. Furthermore,no element, component, or method step in the present disclosure isintended to be dedicated to the public regardless of whether theelement, component, or method step is explicitly recited in the claims.No claim element herein is to be construed under the provisions of 35U.S.C. 112, sixth paragraph, unless the element is expressly recitedusing the phrase “means for.” TABLE 1 HTS Wire radius Copper layerInterval L Embodiment r₁ cm thickness cm meters Tm deg K 1 0.8 0.05 12.515.1 2 0.2 0.1 12.5 7.4 3 0.2 0.1 50 28.2

TABLE 2 Copper layer Tm Heat Load Embodiment thickness cm Coolant deg K(W) 4 0.1 He 130.4* 10 5 0.2 He 55.7 15 6 0.3 He 39.8 20 7 0.1 H₂ 157.1*9 8 0.2 H₂ 87.5 14 9 0.3 H₂ 62.9 18

TABLE 3 Copper layer Tm Heat Load Embodiment thickness cm Coolant deg K(W) 10 0.1 N₂ 93.2 1.2 11 0.2 N₂ 86.2 1.5 12 0.3 N₂ 84.1 1.9

TABLE 4 Interval L Time Embodiment meters Coolant Wire temp K (hrs) 1310 H₂ 96.6 26 14 15 H₂ 97.2 58 15 10 N₂ 94.2 72

1. A conduction-cooled superconducting power transmission cable system,comprising: a wire comprising a high-temperature superconductingmaterial; an inner layer of thermal insulation surrounding said wire; alayer of heat conducting material surrounding said inner layer ofthermal insulation; an outer layer of insulation surrounding said layerof heat conducting material; and means for cooling said wire at or belowits superconducting transition temperature, said means coupled to saidlayer of heat conducting material.
 2. A cable system as recited in claim1, wherein said superconducting wire comprises a material having atransition temperature at or below approximately 110° K.
 3. A cablesystem as recited in claim 1, wherein said wire comprises asuperconducting material chosen from the group consisting of(Bi,Pb)₂Sr₂Ca₂Cu₃O₁₀, YBa₂Cu₃O₇ and MgB₂.
 4. A cable system as recitedin claim 1, wherein said heat conducting material comprises a materialchosen from the group consisting of copper and copper alloys.
 5. A cablesystem as recited in claim 1, wherein said means for cooling comprises acryogenic coolant source.
 6. A cable system as recited in claim 5,wherein said cryogenic coolant source is adapted to use a coolant chosenfrom the group consisting of helium, hydrogen, nitrogen, argon, neon,air and oxygen.
 7. A cable system as recited in claim 5, wherein saidcryogenic coolant source is a stand alone cryogenic compressor system.8. A cable system as recited in claim 5, wherein said cryogenic coolantsource is a modular cryogenic refrigeration system.
 9. A cable system asrecited in claim 5, wherein said cryogenic coolant source is adapted tocool said cable system at length intervals of about 10 meters to atleast about 200 meters.
 10. A cable system as recited in claim 5,wherein said cryogenic coolant source is adapted to cool said cablesystem at length intervals of about 1 meter to at least about 10 meters.11. A cable system as recited in claim 4, wherein said cryogenic coolantsource is adapted to cool said cable system at length intervals of aboutless than 1 meter.
 12. A cable system as recited in claim 11, whereinsaid cryogenic coolant source further comprises a microelectromechanicalsystem (MEMS).
 13. A cable system as recited in claim 11, wherein saidcryogenic coolant source further comprises a nanoelectromechanicalsystem (NEMS).
 14. A cable system as recited in claim 5, wherein saidcryogenic coolant source is adapted to use electricity sourced from saidcable system or from an external source or both.
 15. A cable system asrecited in claim 1, wherein said outer thermal insulation comprises oneor more layers of a highly reflective radiant barrier materialalternating with one or more layers of a low-conductivity spacermaterial.
 16. A cable system as recited in claim 15, wherein said highlyreflective radiant barrier material comprises aluminum foil.
 17. A cablesystem as recited in claim 15, wherein said low conductivity spacermaterial comprises fiberglass paper.
 18. A cable system as recited inclaim 15, wherein said outer thermal insulation comprises a thermalconductivity of about 3.7×10⁻⁵ W/m-K and a layer density of about 20layer/cm.
 19. A cable system as recited in claim 1, wherein the radiusof said wire is at least about 0.1 centimeter to about 1 centimeter. 20.A cable system as recited in claim 1, wherein the radius of said wire isless than or about 0.1 centimeter.
 21. A cable system as recited inclaim 1, wherein the cross section of said wire is rectangular.
 22. Acable system as recited in claim 1, wherein the outer radius of saidinner thermal insulation is at least about 2 centimeters.
 23. A cablesystem as recited in claim 1, wherein the outer radius of said outerthermal barrier is at least about 3 centimeters.
 24. A cable system asrecited in claim 1, wherein the diameter of said cable system is atleast about 20 centimeters.
 25. A cable system as recited in claim 1,wherein said means for cooling is coupled to said wire.
 26. A cablesystem as recited in claim 1, further comprising: a second layer of heatconducting material surrounding said outer layer of thermal insulation;and a second outer layer of insulation surrounding said second layer ofheat conducting material.
 27. A conduction-cooled superconducting cablesystem, comprising: a wire comprising of a high-temperaturesuperconducting material; an inner layer of thermal insulationsurrounding said wire; a plurality of heat conducting material layerssurrounding said inner layer of thermal insulation; a plurality ofinsulation layers surrounding said plurality of heat conducting materiallayers; and means for cooling said wire at or below its superconductingtransition temperature, said means for cooling coupled to said layers ofheat conducting material at periodic intervals along the superconductingcable system.
 28. A cable system as recited in claim 27, wherein saidwire comprises a superconducting material chosen from the groupconsisting of (Bi,Pb)₂Sr₂Ca₂Cu₃O₁₀, YBa₂Cu₃O₇ and MgB₂.
 29. A cablesystem as recited in claim 27, wherein said heat conducting materialcomprises copper or copper alloys.
 30. A cable system as recited inclaim 27, wherein said means for cooling comprises a plurality ofcryogenic coolant sources.
 31. A cable system as recited in claim 30,wherein said cryogenic coolant sources are adapted to use a coolantchosen from the group consisting of helium, hydrogen, nitrogen, argon,neon, air and oxygen.
 32. A cable system as recited in claim 30, whereinsaid cryogenic coolant sources are adapted to cool said layers of heatconducting material at periodic intervals of at least about 1 meter andup to about 200 meters.
 33. A cable system as recited in claim 30,wherein said cryogenic coolant sources are adapted to cool said cablesystem at length intervals of about less than 1 meter.
 34. A cablesystem as recited in claim 30, wherein said cryogenic coolant sourcesfurther comprise a plurality of microelectromechanical systems (MEMS).35. A cable system as recited in claim 30, wherein said cryogeniccoolant sources further comprise a plurality of nanoelectromechanicalsystems (NEMS).
 36. A cable system as recited in claim 27, wherein saidouter thermal insulation comprises one or more layers of a highlyreflective radiant barrier material alternating with one or more layersof a low-conductivity spacer material.
 37. A cable system as recited inclaim 36, wherein said outer thermal insulation comprises a thermalconductivity of about 3.7×10⁻⁵ W/m-K and a layer density of about 20layer/cm.
 38. A method for fabricating a conduction-cooledhigh-temperature superconducting cable comprising: manufacturing a hightemperature superconducting wire; wrapping said wire with an innerthermal barrier; encasing said inner thermal barrier with a layer ofthermal conducting material; wrapping said layer of thermal conductingmaterial with an outer thermal barrier; and coupling a plurality ofcryogenic coolant sources to said layer of thermal conducting materialat predetermined positions.
 39. A method as recited in claim 38, whereinsaid superconducting wire comprises a material having a transitiontemperature below approximately 110° K.
 40. A method as recited in claim38, wherein said wire comprises a superconducting material chosen fromthe group consisting of (Bi,Pb)₂Sr₂Ca₂Cu₃O₁₀, YBa₂Cu₃O₇ and MgB₂.
 41. Amethod as recited in claim 38, wherein said heat conducting materialcomprises copper or copper alloys.
 42. A method as recited in claim 38,wherein said outer thermal insulation comprises layers of a highlyreflective radiant barrier material alternating with a low-conductivityspacer material.
 43. A method as recited in claim 42, wherein saidhighly reflective radiant barrier material comprises aluminum foil andsaid low conductivity spacer material comprises fiberglass paper.
 44. Amethod as recited in claim 42, wherein said outer thermal insulationcomprises a thermal conductivity of about 3.7×10⁻⁵ W/m-K and a layerdensity of about 20 layer/cm.
 45. A method as recited in claim 38,wherein said cryogenic coolant sources are adapted to use a coolantchosen from the group consisting of helium, hydrogen, nitrogen, argon,neon, air and oxygen.
 46. A method as recited in claim 45, wherein saidcryogenic coolant sources are adapted to cool said layers of heatconducting material at periodic intervals along said superconductingcable of at least about 1 meter and up to about 200 meters.
 47. A methodas recited in claim 45, wherein said cryogenic coolant sources areadapted to cool said cable system at periodic intervals along saidsuperconducting cable of about less than 1 meter.
 48. A method asrecited in claim 45, wherein said cryogenic coolant sources furthercomprise a plurality of microelectromechanical systems (MEMS).
 49. Amethod as recited in claim 45, wherein said cryogenic coolant sourcesfurther comprise a plurality of nanoelectromechanical systems (NEMS).50. A method for fabricating a conduction-cooled high-temperaturesuperconducting cable comprising: performing heat transfer calculationsto determine the dimensions of a high temperature superconducting wire,an inner thermal barrier, a layer of thermal conducting material, anouter thermal barrier, and spacing intervals of cryogenic coolantsources necessary to operate the superconducting cable below thecritical temperature of said high temperature superconducting wire;manufacturing said high temperature superconducting wire; wrapping saidwire with said inner thermal barrier; encasing said inner thermalbarrier with said layer of thermal conducting material; wrapping saidlayer of thermal conducting material with said outer thermal barrier;and coupling a plurality of said cryogenic coolant sources to said layerof thermal conducting material at calculated spacing intervals.